Here \(X=V-E\)  as above, the indices of \(X\)  and semicolon denote the covariant differentiation with respect to the Levi-Civita connection \(\tilde{\Gamma }^\rho _{\mu \nu }\) , and the approximate equality means that the relevant part of the expression is retained. Therefore we have (see also [1]})
\(\begin{array}{l}\tilde{a}_4=\frac{1}{4!}\Big [\frac{2}{5}\tilde{R}_{\mu \nu }\tilde{\nabla }^\mu E \tilde{\nabla }^\nu E+\frac{4}{15}\tilde{\nabla }^\mu \Omega _{\mu \nu }\Omega ^{\nu \rho }\tilde{\nabla }_\rho E+\frac{4}{5}\tilde{\nabla }^\mu E\tilde{\nabla }_\mu \tilde{\nabla }^2E-\frac{2}{5}\tilde{\nabla }^\mu E\tilde{\nabla }_\mu \Omega ^2+\frac{1}{3}(\tilde{\nabla }^2E)^2-\\-\frac{1}{3}\tilde{\nabla }^2E\Omega ^2+\frac{1}{12}(\Omega ^2)^2+\frac{4}{15}\tilde{\nabla }^\mu \tilde{\nabla }^\nu E\tilde{\nabla }_\mu \tilde{\nabla }_\nu E+\frac{4}{15}\tilde{\nabla }^\mu \tilde{\nabla }^\nu E\Omega ^2_{\mu \nu }+\frac{1}{15}\operatorname{Tr}\Omega ^4\Big ],\end{array}\) 
